Unravel the geometry and topology behind noisy networks Minghao Tian Abstract Graphs and network data are ubiquitous across a wide spectrum of scientific and application domains. An interesting and important observation that arises very often in real networks is the so-called small- world effect, which is often referred to as the “six degrees of separation” in popular culture. It basically says that the average shortest-path distance between vertex pairs is very short. Many beautiful generative models for graphs have been proposed, partly aiming to understand this small-world effect observed from real networks. Inspired by the celebrated small-world network model proposed by Watts and Strogatz, a variety of follow-up work consider the model that an observed network is obtained by adding random perturbation to a specific type of underlying “structured graph” (such as a grid or a ring). In this proposal, we advocate the perspective that an observed graph is often a noisy version of some discretized 1-skeleton of a hidden domain. We aim to anaylze two aspects of this type of model — geometry and topology. Specifically, the geo- metric problem we aim to solve is to recover the metric structure of the hidden domain from the observed graph, which is orthogonal to the usual studies of network models (which often focuses on characteriz- ing / predicting behaviors and properties of real-world networks). We will consider the following natural network model (called ER-perturbed random geometric graphs or noisy random geometric graphs): We ∗ assume that there is a true graph GX which is a certain proximity graph for points sampled from a hidden ∗ domain X ; while the observed graph GbX is an Erdos–R˝ enyi´ type perturbed version of GX . Two methods are proposed in this research proposal to recover the metric structure of X from GbX : Jaccard-filtering process, which based on Jaccard (similarity) index, and clique-filtering process, which based on edge clique number (a local version of the clique number). We show that these two simple filtering processes can recover this metric within a constant multiplicative factor under our network model. We also consider global and local topological features of the observed graph. We first focus on the clique number of the ER-perturbed random geometric graph GbX , which is an important global graph quantity in both network analysis and graph theory. We provide asymptotic tight bounds of the clique number of GbX under different common settings. Then, we take a refined view of the noisy graphs. loc Specifically, we focus on two types of local subgraphs — neighborhood subgraphs Gu;v for any edge (u; v) in G, which defined as the induced subgraph over the common neighbors of u and v in G, and k rooted (k-neighborhood) subgraphs Gu for any vertex u in G, which is the induced subgraph over the vertices within k distance (shortest-path distance) away from u. We show that the edge clique number in GbX presents two fundamentally different types of behaviors, depending on which “type” of randomness it is generated from. Also, we notice that many graph representations proposed recently are based on rooted subgraphs (or similar substrutures), which may be later used in tasks like network comparison and network classification. However, as we know, the theoretical understanding of the topology of these subgraphs is rather limited. We take a first step to explore the topological features of the rooted subgraphs in Erdos–R˝ enyi´ random graph G(n; p). Specifically, we show that the 1-dimensional Betti number of 1- ring subgraphs (the induced subgraph over vertices exactly 1 distance away from the randomly selected root vertex) satisfies a central limit theorem. 1 Introduction Graphs and networks are ubiquitous across a wide spectrum of scientific and application domains. Analyzing various types of graphs and network data play a fundamental role in modern data science. In the past 1 several decades, there has been a large amount of research studying various aspects of graphs, ranging from developing efficient algorithms to process graphs, to graph-based data mining. Among them a varity of empirical studies focus on the graph properties (such as the degree distribution and the clustering coefficient) of different types of real networks. An interesting and important observation that arises very often and has practical implications is the celebrated small-world effect [41] discovered in a seminal work of Milgram [37], which is often referred to as the “six degrees of separation” in popular culture. It basically says that the average shortest-path distance, appropriately defined, between vertex pairs is very short. Although first studied in friendship networks, this phenomenon appears to be occurring in almost all types of networks. Many beautiful generative models for graphs have been proposed, partly aiming to understand this small- world effect observed from real networks [41, 56]. One of the most classic models with theoretical guarantee is the Erdos–R˝ enyi´ random graph model G(n; p) [16, 17], constructed by adding edges between all pairs of n vertices with probability p independently. However, since this model is purely combinatorial, it fails to capture the geometry (shape) of the network. For example, most people make friends based on common interests, location, age and so on. In other words, in friendship networks, vertices could be sampled from some feature space of people, and two people could be connected if they are nearby in the feature space. Obviously, the structure of the feature space cannot be encoded in the Erdos–R˝ enyi´ random graph model G(n; p). Another line of such generative graph models assumes that an observed network is obtained by adding random perturbation to a specific type of underlying “structured graph” (such as a grid or a ring). For example, the much-celebrated model introduced by Watts and Strogatz [56] generates a network by starting with a k-nearest neighbor graph spanned by vertices regularly distributed along a ring. It then randomly “rewires” some of the edges connecting neighboring points to instead connect nodes possibly far away. Watts and Strogatz showed that this simple model displays two important characteristics seen in small-world networks: low diameter in shortest path metric and high clustering coefficients. There have since been many variants of this model proposed so as to generate small-world networks, such as adding random edges in a distance-dependent manner [28, 50], or extending similar ideas to incorporate hierarchical structures in networks; e.g, [29, 55]. See [30] for a survey on this topic. Statement of the problems. Inspired by the small-world model by Watts and Strogatz (and some later variants), we take the perspective that an observed graph can be deemed as a noisy snapshot of the graph representation (discretized 1-skeleton) of a hidden domain of interest. However, orthogonal to the usual studies of this type of network models (which often focuses on simulating real-world networks and inter- preting the observed phenomena), we aim to answer the following two main questions: 1. (Geometry) What can we infer about the hidden domain from the observed graph? 2. (Topology) What are the topological properties (such as the clique number) of the observed graph? To be more specific, we propose the following network model in [45]: Assume that the hidden space that generates data is a “nice” probability measure µ supported on a compact metric space X = (X; dX ) (e.g, the uniform probability measure supported on an embedded smooth low-dimensional Riemannian manifold). Suppose that the data points V are sampled i.i.d from this probability measure µ, and the “true graph” ∗ Gr connecting them is the r-neighborhood graph spanned by V (i.e, two points u; v are connected if their distance dX (u; v) ≤ r). The observed graph Gb however is only a noisy version of the true proximity graph ∗ ∗ Gr, and we model this noise by an Erdos-R¨ enyi´ (ER) type perturbation – each edge in the true graph Gr can be deleted with probability p, while a “short-cut” edge between two unconnected vertices u; v could be ∗ inserted to Gr with probability q. This model is later called the ER-perturbed random geometric graph [25] 2 ∗ or the noisy random geometric graph [26], as the true proximity graph Gr generated in this way is in fact a random geometric graph in random graph theory [46]. To motivate this model, imagine in a social network a person typically makes friends with other persons that are close to herself in the unknown feature space modeled by our metric space X . The distribution of people (graph vertices) is captured by the measure µ on X . However, there are always (or may be even many) exceptions – friends could be established by chance, and two seemingly similar persons (say, close geographically and in tastes) may not develop friendship. Thus it is reasonable to model an observed social ∗ network Gb as an ER-type perturbation of the proximity graph Gr to account for such exceptions. Assumptions and notations. All the graphs mentioned in this proposal are simple undirected graphs, which means there are no duplicate edges or loops (a loop is an edge that connects a vertex to itself). For any graph G, let V (G) and E(G) refer to its vertex set and edge set, and let NG(u) denote the set of neighbors of vertex u in G (i.e. vertices connected to u 2 V (G) by edges in E(G)). We also use the terminology with high probability. If A1;A2; ··· is a sequence of events, then “An happens with high probability” means that limn!1 P[An] = 1 + o(1).